Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001820
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([(0,1)],2)
 => 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2}}
 => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2,3}}
 => [2,3,1] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,2,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [4,6,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [3,6,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [5,2,6,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [3,5,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
{{1,2,5},{3},{4},{6}}
 => [2,5,3,4,1,6] => [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
{{1,2},{3,5,6},{4}}
 => [2,1,5,4,6,3] => [4,1,3,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => [5,1,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [6,3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [3,4,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => [5,1,3,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => [2,5,3,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
Description
The size of the image of the pop stack sorting operator. The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St000772
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => ? = 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? = 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => ? = 2
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? = 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 2
{{1,3,4},{2}}
 => [3,2,4,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? = 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? = 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [5,1,2,6,4,3] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [3,6,4,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => 1
{{1,2,5},{3},{4},{6}}
 => [2,5,3,4,1,6] => [4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 1
{{1,2},{3,5,6},{4}}
 => [2,1,5,4,6,3] => [2,1,6,3,5,4] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ? = 1
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [5,1,2,4,6,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ? = 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [6,2,4,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [5,1,3,2,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => [2,6,1,4,5,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
{{1,5},{2,3,6},{4}}
 => [5,3,6,4,1,2] => [2,6,4,1,5,3] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
{{1,5},{2,3},{4,6}}
 => [5,3,2,6,1,4] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
{{1},{2,3,5},{4,6}}
 => [1,3,5,6,2,4] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1
{{1},{2,3},{4,5,6}}
 => [1,3,2,5,6,4] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ? = 1
{{1},{2,3,6},{4},{5}}
 => [1,3,6,4,5,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 1
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
{{1,4,6},{2},{3,5}}
 => [4,2,5,6,3,1] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,4},{2},{3,5},{6}}
 => [4,2,5,1,3,6] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1},{2,4},{3,5,6}}
 => [1,4,5,2,6,3] => [1,6,3,5,2,4] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1},{2,4},{3,5},{6}}
 => [1,4,5,2,3,6] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
 => ? = 1
{{1},{2,4,6},{3},{5}}
 => [1,4,3,6,5,2] => [1,6,5,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 1
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
{{1,5},{2},{3},{4,6}}
 => [5,2,3,6,1,4] => [4,6,1,3,5,2] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1},{2,5},{3,6},{4}}
 => [1,5,6,4,2,3] => [1,3,6,4,2,5] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1},{2},{3,5},{4,6}}
 => [1,2,5,6,3,4] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ? = 1
{{1,2,4,5,6},{3,7}}
 => [2,4,7,5,6,1,3] => [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 1
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => [7,3,5,1,2,4,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,2,4,7},{3},{5},{6}}
 => [2,4,3,7,5,6,1] => [6,1,2,4,3,7,5] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,2,4},{3,7},{5},{6}}
 => [2,4,7,1,5,6,3] => [6,3,5,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [6,5,1,2,7,4,3] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,2},{3,4,7},{5},{6}}
 => [2,1,4,7,5,6,3] => [2,1,6,3,4,7,5] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 1
{{1,2,5,6},{3,7},{4}}
 => [2,5,7,4,6,1,3] => [3,6,1,2,7,4,5] => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => 1
{{1,2,5},{3,7},{4},{6}}
 => [2,5,7,4,1,6,3] => [6,3,7,4,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
{{1,2},{3,5,6,7},{4}}
 => [2,1,5,4,6,7,3] => [2,1,7,3,5,4,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [2,1,7,4,6,3,5] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1,2},{3,5,7},{4},{6}}
 => [2,1,5,4,7,6,3] => [2,1,7,6,3,5,4] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1,2},{3,6},{4,5,7}}
 => [2,1,6,5,7,3,4] => [2,1,4,7,3,6,5] => ([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1,2,7},{3},{4,5,6}}
 => [2,7,3,5,6,4,1] => [6,4,1,2,5,7,3] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,2,6},{3,7},{4},{5}}
 => [2,6,7,4,5,1,3] => [3,5,1,2,7,4,6] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => 1
{{1,2},{3,6},{4,7},{5}}
 => [2,1,6,7,5,3,4] => [2,1,4,7,5,3,6] => ([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1,2},{3,6},{4},{5,7}}
 => [2,1,6,4,7,3,5] => [2,1,5,7,3,6,4] => ([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1,3,4,5,6},{2,7}}
 => [3,7,4,5,6,1,2] => [2,6,1,3,5,7,4] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,3,4},{2,6,7},{5}}
 => [3,6,4,1,5,7,2] => [7,2,5,6,4,1,3] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,3,4},{2,6},{5,7}}
 => [3,6,4,1,7,2,5] => [5,7,2,6,4,1,3] => ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,3,4},{2,7},{5},{6}}
 => [3,7,4,1,5,6,2] => [6,2,5,7,4,1,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1
{{1,3,5,6},{2,7},{4}}
 => [3,7,5,4,6,1,2] => [2,6,1,3,7,5,4] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1
{{1,3,6},{2,5},{4,7}}
 => [3,5,6,7,2,1,4] => [4,7,2,1,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
{{1,3,7},{2},{4,5,6}}
 => [3,2,7,5,6,4,1] => [6,4,1,3,2,7,5] => ([(0,2),(1,5),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
{{1,3},{2,7},{4,5,6}}
 => [3,7,1,5,6,4,2] => [6,4,2,5,7,1,3] => ([(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
 => 1
{{1,3,7},{2},{4,5},{6}}
 => [3,2,7,5,4,6,1] => [6,1,3,2,7,5,4] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1
{{1},{2,3,5},{4,6,7}}
 => [1,3,5,6,2,7,4] => [1,7,4,6,2,3,5] => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1},{2,3,5,7},{4},{6}}
 => [1,3,5,4,7,6,2] => [1,7,6,2,3,5,4] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1},{2,3,6},{4,7},{5}}
 => [1,3,6,7,5,2,4] => [1,4,7,5,2,3,6] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1
{{1},{2,3},{4,6,7},{5}}
 => [1,3,2,6,5,7,4] => [1,3,2,7,4,6,5] => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
{{1},{2,3},{4,6},{5,7}}
 => [1,3,2,6,7,4,5] => [1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
 => ? = 1
{{1},{2,4,5,6},{3,7}}
 => [1,4,7,5,6,2,3] => [1,3,6,2,4,7,5] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 1
{{1},{2,4},{3,5,6,7}}
 => [1,4,5,2,6,7,3] => [1,7,3,5,2,4,6] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].